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Math Help - Moment Generating function

  1. #1
    Junior Member
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    Moment Generating function

    Hi,

    I'm just need to work out a step taken in the answer to the following question:

    A random variable U has the uniform distribution on the range (0, 1). Find the moment
    generating function of W, where W is given by W = − ln U, and hence identify the
    distribution of W.

    Hint: The moment generating function, with argument t, of the exponential distribution with mean θ is (1 − θt)^(−1), for t < θ^(−1)

    Now the answer goes like this:

    Moment Generating Function = E(exp(tW))
    =E(exp(t(-logU)))
    =E((exp(logU))^(-t))
    =E(U^(-t))

    Now from here i don't fully understand how they get to the next step:

    =int(1*u^(-t),u=0..1)
    and from here they go on to show
    Moment Generating Function = 1/(1-t)

    I understand how to find expectancies, by integrating, but i am unsure why it is 1*, if someone could explain this to me i would be very greatful.

    p.s sorry for not using Latex but it doesn't seem to be working.
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  2. #2
    MHF Contributor matheagle's Avatar
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    the density of f is 1 on (0,1)

    E(U^{-t})=\int_0^1 u^{-t} f(u)du where f(u)=1

    then just use calc 1 \int_0^1 u^{-t} du =u^{-t+1}/(-t+1)
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